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Standing wave ratio

The Standing wave ratio is often measured in terms of the Voltage Standing Wave Ratio(VSWR). Put simply, this is the ratio of the maximum to the minimum voltage of astanding wave (which is the instantaneous sum of incident and reflected waves). Ideal is a

figure of 1:1 which means that 100% of the incoming signal passed through thecomponent without any reflection. In that case, there would be no standing wave. A 2:1VSWR (or mismatch) means that 12% of the incoming signal was reflected.

In telecommunications, standing wave ratio (SWR ) is the ratio of the amplitude of a partial standing wave at an antinode (maximum) to the amplitude at an adjacent node (minimum).

The SWR is usually defined as a voltage ratio called the VSWR, for voltage standing wave ratio . It is also possible to define the SWR in terms of current , resulting in theISWR, which has the same numerical value. The power standing wave ratio (PSWR) is

defined as the square of the SWR.

The voltage component of a standing wave in a uniform transmission line consists of theforward wave (with amplitude V f ) superimposed on the reflected wave (with amplitudeV r ).

Reflections occur as a result of discontinuities, such as an imperfection in an otherwiseuniform transmission line, or when a transmission line is terminated with other than itscharacteristic impedance . The reflection coefficient Γ is defined thus:

Γ is a complex number that describes both the magnitude and the phase shift of thereflection. The simplest cases, when the imaginary part of Γ is zero, are:

• Γ = − 1: maximum negative reflection, when the line is short-circuited,• Γ = 0: no reflection, when the line is perfectly matched,• Γ = + 1: maximum positive reflection, when the line is open-circuited.

For the calculation of VSWR, only the magnitude of Γ, denoted by ρ, is of interest.

At some points along the line the two waves interfere constructively, and the resultingamplitude V max is the sum of their amplitudes:

At other points, the waves interfere destructively, and the resulting amplitude V min is thedifference between their amplitudes:

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The voltage standing wave ratio is then equal to:

As ρ, the magnitude of Γ, always falls in the range [0,1], the VSWR is always ≥ +1.

The SWR can also be defined as the ratio of the maximum amplitude of the electric field strength to its minimum amplitude, i.e. E max / E min.

Further analysis

To understand the standing wave ratio in detail, we need to calculate the voltage (or,equivalently, the electrical field strength) at any point along the transmission line at anymoment in time. We can begin with the forward wave, whose voltage as a function of time t and of distance x along the transmission line is:

where A is the amplitude of the forward wave, ω is its angular frequency and k is aconstant (equal to ω divided by the speed of the wave). The voltage of the reflected waveis a similar function, but spatially reversed (the sign of x is inverted) and attenuated bythe reflection coefficient ρ:

The total voltage V t on the transmission line is given by the principle of superposition ,which is just a matter of adding the two waves:

Using standard trigonometric identities, this equation can be converted to the followingform:

where

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This form of the equation shows, if we ignore some of the details, that the maximumvoltage over time V mot at a distance x from the transmitter is the periodic function

This varies with x from a minimum of A(1 − ρ) to a maximum of A(1 + ρ), as we saw inthe earlier, simplified discussion. A graph of V mot against x, in the case when ρ = 0.5, isshown below. V min and V max are the values used to calculate the SWR.

Standing wave ratio for ρ = 0.5

It is important to note that this graph does not show the instantaneous voltage profilealong the transmission line. It only shows the maximum amplitude of the oscillation ateach point. The instantaneous voltage is a function of both time and distance, so couldonly be shown fully by a three-dimensional or animated graph.

Practical implications of SWR

SWR has a number of implications that are directly applicable to radio use.

1. SWR is an indicator of reflected waves bouncing back and forth within thetransmission line, and as such, an increase in SWR corresponds to an increase in

power in the line beyond the actual transmitted power. This increased power willincrease RF losses, as increased voltage increases dielectric losses, and increasedcurrent increases resistive losses.

2. Matched impedances give ideal power transfer; mismatched impedances givehigh SWR and reduced power transfer.

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3. Higher power in the transmission line also leaks back into the radio, which causesit to heat up.

4. The higher voltages associated with a sufficiently high SWR could damage thetransmitter. solid state radios which have a lower tolerance for high voltages mayautomatically reduce output power to prevent damage. Tube radios may arc. The

high voltages may also cause transmission line dielectric to break down and/or burn.5. VSWR measurements may be taken to ensure that a waveguide is contiguous and

has no leaks or sharp bends. If such bends or holes are present in the waveguidesurface, they may diminish the performance of both TX and RX equipmentstrings. Arcing may occur if there is a hole, if transmitting at high power, usually200 watts or more (Need reference for the power statement). Waveguide

plumbing is crucial for proper waveguide performance. Reflected power mayoccur and damage equipment as well. Another cause of bad VSWR in awaveguide is moisture build-up, which can typically be prevented with silica gel .

The Effects of VSWR on Transmitted PowerBy James G. Lee, W6VAT

No matter how long you have been a ham, sooner of later you will be involved in at leastone discussion of something called the Voltage Standing Wave Ratio, or VSWR, of anantenna system. There is a lot of good information available on VSWR as well as a lotmisconception about what it is and what it signifies. Probably the most oftenmisconception is that your VSWR should be as close to 1:1 as possible; otherwise “youwon't get out very well." A 1:1 VSWR implies a perfect match between all elements of the antenna system. The only problem is that it is possible to have a low VSWR and stillhave some very serious things wrong with your antenna system. Other misconceptionssuch as a high VSWR causing television interference, or other unwanted problems areoften heard and can cause unnecessary worry. The concept of VSWR is easy to grasp andits importance in an antenna system does not require an engineering degree to understand.

WHY VSWR EXISTS

Early in electronics you learned that to get maximum power into a load required that theload impedance matches the generator impedance. Any difference, or mismatching, of these impedance would not produce maximum power transfer. This is true of antennasand transmitters as well but, except for handie-talkies, most antennas are not connected

directly to a transmitter. The antenna is usually located some difference from thetransmitter and requires a feedline to transfer power between the two. If the feedline hasno loss, and matches both the transmitter output impedance and the antenna inputimpedance, then - and only - then will maximum power be delivered to the antenna. Inthis case the VSWR will be 1:1 and the voltage and current will be constant over thewhole length of the feedline. Any deviation from this situation will cause a "standingwave" of voltage and current to exist on the line.

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There are a number of ways VSWR or its effects can be described and measured.Different terms such as reflection coefficient, return loss, reflected power, and transmitted

power losses are but a few. They are not difficult concepts to understand, since in mostinstances there are different ways of saying the same thing. The proportion of incident (or forward) power which is reflected back toward the transmitter by a mismatched antenna

is called reflected power and is determined by the reflection coefficient at the antenna.The reflection coefficient "p" is simply a measure of this mismatch seen at the antenna bythe feedline and is equal to:

P =(Z1-Z o)/(Z 1+Z o)

Here Z 1 is the antenna impedance and Z o is the feedline impedance. Both Z 1 and Z o arecomplex numbers so "p" is also a complex number.

You remember from elementary AC mathematics that a complex number has a "phaseangle" associated with it. The phase of the reflected signal will be advanced or delayed

depending upon whether the antenna appears inductive or capacitive to the feedline. If theantenna appears inductive the voltage will be advanced in phase, and if the antenna iscapacitive, the voltage will be retarded. The reflective signal travels back to thetransmitter and adds to the incident signal at that point.

Thus, any mismatch at the antenna gives rise to a second 'travelling wave' which goes inthe opposite direction from the incident wave. When Z 1 = Z o the reflection coefficient iszero and there is no reflected signal. In this case all power is accepted by the antenna andthis is the ideal situation where VSWR is concerned. The problem is that this condition israrely, if ever, achieved and so "p" will have a value different from zero. Note that "p"can have negative values, but in calculating VSWR from the reflection coefficient, only

the "absolute value" is used - which is a positive value laying between 0 and 1.As the two traveling waves pass each other in opposite directions, they set up aninterference pattern called a "standing wave". At certain places on the feedline thevoltages will add producing a voltage maximum, and at others their relative phasedifference will cause a voltage minimum to exist on the feedline. These maximum andminimum points occur 1/4 wavelength apart. In the days when open-wire feedlines wereused these points could easily be measured with simple indicators. Coax cable however

presents another problem since the "inside" of the cable is not readily available for measurements. Consequently, VSWR measurements on coax are usually made at thetransmitter end of the feedline. Therefore you are presented with the VSWR of the entiresystem which includes all losses associated with the entire system.

INTERPRETING WHAT YOU HAVE READ

Many VSWR meters are calibrated to read FORWARD power as well as REFLECTED power. They may actually be measuring voltage, and simply have the scales calibrated in power. The important point is to understand what the meter is actually telling you.Assuming for the moment that the VSWR meter contributes no errors, the FORWARD

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reading is the SUM of the forward power and the reflected power. As a result, it is greater than your actual power output. The REFLECTED power reading is that amount of power which was not initially absorbed by the antenna and has been sent back down thefeedline. At the transmitter end it encounters the transmitter output circuitry and is re-reflected back towards the antenna. This happens because you do, in fact, have a VSWR

greater than 1:1 as seen by the transmitter. When the re-reflected power encounters theantenna, a portion of it is absorbed and the whole process starts over again.

Ultimately then, most of your signal is eventually absorbed by the antenna. You might betempted to think that all of this bouncing back and forth would cause "smearing or

blurring” of your signal but this is not so. The average transmitted signal appears as a"steady-state" signal to the feedline and antenna. Remember your signal is travelling at asignificant fraction of the speed of light. For instance, the velocity of propagation of RG-8/A is 0.66 or 2/3 the speed of light. The speed of light is close to 1000 feet per microsecond, and a dot or voice peak takes milliseconds to complete. If the speed of lightwere 20 miles-per-hour then the situation would be completely different and we probably

wouldn't have radio transmission at all. (Ed. Note, it would be as fast as the mail then.)

Given the reality then that almost all power launched down a feedline reaches andabsorbed by the antenna, one has to wonder why VSWR is all that important. Theimportance is due to the fact that feedlines have losses and, antennas have somethingcalled radiation efficiency. They are what make proper interpretation of VSWR important. Power is lost due to feedline attenuation and this loss goes up as the VSWR goes up. The efficiency of an antenna is determined by the ratio of its "radiationresistance" to its "loss resistance". Antenna efficiency can simply described by thefollowing equation:

% Efficiency=[R a/(R a+R loss)] X 100The radiation resistance is R a, and R loss is made up of any associated losses of the antennasuch as loading coils and ground systems. How well you "get out" therefore dependsmore on low losses and efficient antennas than on what your actual VSWR is as thefollowing example will show.

THE EFFECTS OF ATTENUATION ON VSWR

Early in this discussion the statement was made that your VSWR may appear to be verylow and yet there could be serious things wrong with your antenna system. Figure 1

shows how this can happen. The curves in the figure represent the forward and thereflected voltage on an antenna which has a feedline loss of 3 dB. and a reflectioncoefficient of p=0.5. In this example the actual value of voltage is inconsequential andcan be considered to be "E". We are only interested in relative values of "E" in any case.The length of the feedline is also arbitrary since we are only concerned with its total loss

between transmitter and antenna.

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The signal voltage "E" starts out at full value -1.0 E - on the feed line and is attenuated ata 3-dB rate. This means that the voltage will only be 71% - or 0.707E - when it reachesthe antenna terminals. Remember that while 3-dB is a factor of two for power considerations, power is proportional to E-squared, consequently E will be only 0.71ewhen it arrives at the antenna input. The top curve in Figure 1 shows the FORWARDvoltage decay as it travels down the feedline to the antenna input.

Since the antenna in this example has a reflection coefficient of 0.5, this means that 1/2 of the incident voltage will be reflected back down the feedline. This value is (0.5 X o.71E)or 0.35E volts. The feedline has no way of knowing which way signals are traveling, sothis reflected voltage will suffer the same 3-dB attenuation on the return trip. When itarrives back at the transmitter end of the feedline its value is only (0.71 X 0.35E) or 0.25volts. The VSWR meter sees this value and since

VSWR=(E fwd + E ref )/(E fwd - E ref )

The VSWR meter will read 1.67:1

That value of VSWR is guaranteed is to make almost everyone happy, but your antennasystem is not very good. The 3-dB loss down the feedline means only 1/2 of your output

power reaches the antenna, and if your antenna has significant losses, something less than1/2 of your power will be radiated depending upon how bad the losses really are. If for instance, the loss resistance equals your radiation resistance, the antenna is only 50%efficient meaning only 1/4 of your output power is actually radiated. Yet that reading of 1.67:1 looks fine. A reflection coefficient of p =0.5 means your antenna is not well

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matched to the feedline. VSWR can be calculated from the reflection coefficient by thefollowing:

VSWR = (1+p)/(1-p)

Using this formula shows your VSWR at the antenna is 3:1, quite a different value thanyour VSWR meter reads. The difference in the input and output VSWR values is due tothe loss introduced by the feedline. Figure 2 shows how this loss can cause you to get adifferent VSWR depending upon where you measure VSWR in a feedline. You canmeasure VSWR at the antenna end of the feedline, but it is usually impractical to do.

Figure 2

You can use 1/2 wavelengths of coax between your VSWR meter and the antenna because a 1/2 wavelength of cable repeats the impedance it sees. The only problem is thatyou are introducing other possible elements into your measurements. But let's assumethat your VSWR measurement at the feedline is reasonably close to what is actuallyoccurring on the feed line, and that your feedline losses are not great. The burningquestion still is "how good or bad is the VSWR reading?"

VSWR AND TRANSMITTED POWER

Let's assume you have an efficient antenna, fed with a low-loss feedline so that theVSWR meter at the transmitter gives you a true reading of 1.65:1. There is no real reasonto try to lower it, in fact the same would hold true if the reading were 2:1. Figure 3 is achart showing the equivalence of VSWR to RETURN LOSS(dB), REFLECTEDPOWER(%) and TRANSMISSION LOSS(dB). Return loss is related to reflectioncoefficient by the equation:

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Return Loss = -20log 10(p)

It is just another way of measuring VSWR. For example, with a perfect 1:1 VSWR therewould be no reflected power consequently the return loss on the feedline would appear to

be infinite. A short or open circuit at the antenna is the worst case scenario since the

reflection coefficient would be p =1.0. All incident power would be reflected, and with alossless feedline the return loss would be 0-dB. this is what the RETURN LOSS (dB)column refers to

The most informative columns in Figure 3 are the REFLECTED POWER(%) and theTRANSMISSION LOSS(dB) columns since they provide an answer to our question of whether further reduction of VSWR is worthwhile. Figure 3 shows that for a VSWR of 1.65:1 the reflected power is only 6.2% of the incident power, and the transmission loss isonly 0.27 dB. In more familiar terms, if you count an S-unit as 6 dB, then the 0.27 dBloss is only 1/22 of an S-unit. A reduction of the VSWR to 1.5:1 would provide only a0.09 dB reduction in transmission loss. This is not worth the effort it would take to

achieve such a miniscule increase in power.Figure 3

VSWR Return Loss(dB)

ReflectedPower (%)

Transmiss.Loss (dB) VSWR Return Loss

(dB)Reflected

Power (%)Transmiss.Loss (dB)

1.00 oo 0.000 0.000 1.38 15.9 2.55 0.1121.01 46.1 0.005 0.0002 1.39 15.7 2.67 0.1181.02 40.1 0.010 0.0005 1.40 15.55 2.78 0.1221.03 36.6 0.022 0.0011 1.41 15.38 2.90 0.1261.04 34.1 0.040 0.0018 1.42 15.2 3.03 0.1321.05 32.3 0.060 0.0028 1.43 15.03 3.14 0.1371.06 30.7 0.082 0.0039 1.44 14.88 3.28 0.1421.07 29.4 0.116 0.0051 1.45 14.7 3.38 0.1471.08 28.3 0.144 0.0066 1.46 14.6 3.50 0.1521.09 27.3 0.184 0.0083 1.47 14.45 3.62 0.1571.10 26.4 0.228 0.0100 1.48 14.3 3.74 0.1641.11 25.6 0.276 0.0118 1.49 14.16 3.87 0.1721.12 24.9 0.324 0.0139 1.50 14.0 4.00 0.181.13 24.3 0.375 0.0160 1.55 13.3 4.8 0.211.14 23.7 0.426 0.0185 1.60 12.6 5.5 0.241.15 23.1 0.488 0.0205 1.65 12.2 6.2 0.271.16 22.6 0.550 0.0235 1.70 11.7 6.8 0.311.17 22.1 0.615 0.0260 1.75 11.3 7.4 0.341.18 21.6 0.682 0.0285 1.80 10.9 8.2 0.371.19 21.2 0.750 0.0318 1.85 10.5 8.9 0.401.20 20.8 0.816 0.0353 1.90 10.2 9.6 0.441.21 20.4 0.90 0.0391 1.95 09.8 10.2 0.471.22 20.1 0.98 0.0426 2.00 09.5 11.0 0.501.23 19.7 1.08 0.0455 2.10 09.0 12.4 0.57

1.24 19.4 1.15 0.049 2.20 08.6 13.8 0.651.25 19.1 1.23 0.053 2.30 08.2 15.3 0.731.26 18.8 1.34 0.056 2.40 07.7 16.6 0.801.27 18.5 1.43 0.060 2.50 07.3 18.0 0.881.28 18.2 1.52 0.064 2.60 07.0 19.5 0.951.29 17.9 1.62 0.068 2.70 06.7 20.8 1.031.30 17.68 1.71 0.073 2.80 06.5 22.3 1.101.31 17.4 1.81 0.078 2.90 06.2 23.7 1.171.32 17.2 1.91 0.083 3.00 06.0 24.9 1.251.33 17.0 2.02 0.087 3.50 05.1 31.0 1.611.34 16.8 2.13 0.092 4.00 04.4 36.0 1.93

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1.35 16.53 2.23 0.096 4.50 03.9 40.6 2.271.36 16.3 2.33 0.101 5.00 03.5 44.4 2.561.37 16.1 2.44 0.106 6.00 02.9 50.8 3.08

Further examination of the chart shows that a VSWR of 2.6:1 results in only about 1 dBof transmission loss. A high VSWR of 6:1 shows just a 3 dB transmission loss, but this is

1/2 an S-unit. You will still be getting out but this is becoming a significant loss. Your feedline will be dissipating more power than it should, and there may be other seriousthings wrong with your antenna system.

Throughout this article you've noticed the use of the term "antenna system". The word"system" means you must pay attention to other things besides just the VSWR and your

power output. Each component of your antenna system must be optimized to get the bestresults. Many factors must be considered such as operating frequencies, bandwidthrequirements of the antenna system, heights, and directivity, all of which affect itsefficiency. Since the height of your antenna, and your operating frequency determine boththe length of the feedline and its losses the interfaces become very important. So there are

a number of trade-offs which must be considered when you contemplate putting up agood antenna system, but these are tales for other times.

You can build or buy your own VSWR meter, but make sure that you understand what itis measuring and what it is really telling you. Then once you are satisfied that you have

put up a really efficient antenna, fed with a low loss feedline, you can sleep well knowingthat to try to reach the ultimate 1:1 VSWR is only an ego trip. As a rule of thumb, anyaccurate VSWR reading under 2:1 is probably not worth the effort to achieve if the other elements of your antenna system are the best you can make them. In fact you might besurprised to find that you really do have a low VSWR when you put up the best antennaand feedline you can. There is an old saying in ham radio that "a dime in the antenna is

worth a dollar in the transmitter any day". Try it and see if you don't agree. -30-